Problem 19 - Basic Engineering Mathematics

Question

Find the sum of the first 7 terms of the series, \(\frac{1}{2}, 1 \frac{1}{2}, 4 \frac{1}{2}, 13 \frac{1}{2}, \ldots\)

Step-by-Step Solution

Answer

The sum of the first 7 terms is 546.5.

Step 1: Identify the Sequence Type

The given sequence is\[\frac{1}{2},\ 1\frac{1}{2},\ 4\frac{1}{2},\ 13\frac{1}{2},\ \ldots\]To identify the type of sequence, we find the differences between consecutive terms:\[\begin{align*}1\frac{1}{2} - \frac{1}{2} &= 1,\4\frac{1}{2} - 1\frac{1}{2} &= 3,\13\frac{1}{2} - 4\frac{1}{2} &= 9.\end{align*}\]The differences are not constant, so it's not an arithmetic sequence. Let's check the ratio instead by dividing successive terms by the previous term.

Step 2: Determine the Sequence Formula

Calculating the ratios between consecutive terms, we have the following\[\begin{align*}\frac{1.5}{0.5} &= 3,\\frac{4.5}{1.5} &= 3,\\frac{13.5}{4.5} &= 3.\end{align*}\]Each term is 3 times the previous term, which indicates it's a geometric sequence with a common ratio \(r=3\). The first term \(a_1\) is \(\frac{1}{2}\). The formula for the \(n\)-th term of a geometric sequence is \[a_n = a_1 \cdot r^{(n-1)}\].

Step 3: Find the First 7 Terms of the Sequence

To find the first seven terms, use the geometric sequence formula \(a_n = \frac{1}{2} \cdot 3^{(n-1)}\) for each \(n\) from 1 to 7:\[\begin{align*}a_1 &= \frac{1}{2}, \a_2 &= \frac{1}{2} \cdot 3^1 = 1.5, \a_3 &= \frac{1}{2} \cdot 3^2 = 4.5, \a_4 &= \frac{1}{2} \cdot 3^3 = 13.5, \a_5 &= \frac{1}{2} \cdot 3^4 = 40.5, \a_6 &= \frac{1}{2} \cdot 3^5 = 121.5, \a_7 &= \frac{1}{2} \cdot 3^6 = 364.5.\end{align*}\]Totalling these, the elements are \(0.5, 1.5, 4.5, 13.5, 40.5, 121.5, 364.5\).

Step 4: Calculate the Sum of First 7 Terms

The sum \(S_n\) of the first \(n\) terms of a geometric series is given by the formula \[S_n = a_1 \frac{r^n - 1}{r - 1}\]where \(a_1 = 0.5\), \(r = 3\), and \(n = 7\). Plugging these values in:\[S_7 = \frac{1}{2} \cdot \frac{3^7 - 1}{3 - 1} = \frac{1}{2} \cdot \frac{2187 - 1}{2} = \frac{1}{2} \cdot \frac{2186}{2} = \frac{1}{2} \cdot 1093 = 546.5.\]Therefore, the sum of the first 7 terms is 546.5.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key element that defines the relationship between terms in the sequence. To identify it, you need to divide each term by its preceding term. This division gives a constant value if the sequence is indeed geometric. This constant is called the *common ratio*.

For instance, consider the sequence given in the exercise:

• First, the second term (1.5) divided by the first term (0.5) gives 3.
• Second, the third term (4.5) divided by the second term (1.5) also gives 3.
• Lastly, the fourth term (13.5) divided by the third term (4.5) yields 3.

Each of these divisions results in the same common ratio, 3. This verifies our sequence as a geometric sequence with a common ratio of 3. Understanding this uniform multiplicative relation helps in predicting future terms.

Geometric Sequence Formula

Once a geometric sequence is identified through its common ratio, you can determine its terms using the geometric sequence formula. This formula allows you to find any term in the sequence without explicitly knowing all preceding terms. The formula is written as:

• \[ a_n = a_1 \cdot r^{(n-1)} \]
• Here, \( a_n \) is the \( n \)-th term, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

In the given exercise, the first term \( a_1 \) is 0.5 and the common ratio \( r \) is 3. To find any term, say the 5th term \( a_5 \), you would calculate:

• \[ a_5 = 0.5 \cdot 3^{4} = 40.5 \]

This approach is efficient and powerful especially for long sequences, offering a direct path to individual terms without the need for progressive calculations.

Sum of Geometric Series

The sum of a geometric series is computed using a specific formula that accounts for the sum of its terms up to a certain point. The formula is:

• \[ S_n = a_1 \cdot \frac{r^n - 1}{r - 1} \]
• Where \( S_n \) is the sum of the first \( n \) terms, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the total number of terms considered.

Let's apply this to the exercise. For the first 7 terms of the series, with \( a_1 = 0.5 \), \( r = 3 \), and \( n = 7 \):

• \[ S_7 = 0.5 \cdot \frac{3^7 - 1}{3 - 1} \]
• Calculate the powers and operations: \( 3^7 = 2187 \), so \( S_7 = 0.5 \cdot \frac{2187 - 1}{2} = 0.5 \cdot \frac{2186}{2} \).
• Further calculating, you get \( 0.5 \cdot 1093 = 546.5 \).

So, the sum of the first 7 terms of this geometric series is 546.5. This formula is practical for handling large series where manually adding each term is cumbersome.